that the energetic ion (i.e. diffuse ion) partial density falls off exponentially from the shock into the upstream region along the magnetic field. In order to prove that the diffuse ions truly undergo a diffusive transport, the spatial variation of the diffuse ion partial density needs to be determined. Ipavich et al. (1981) analyzed about 30 upstream ion events and found that the differential ion flux for the 33 keV ions falls off exponentially with distance from the shock. Trattner et al. (1994) extended the work by Ipavich et al. (1981) by performing a statistical analysis of ∼330 diffuse ion events in an energy range between 10 and 67 keV and found that the intensity of the upstream ions falls off exponentially with an e-folding distance between 3 and 11.7 Re over the investigated energy range.
These studies indeed demonstrate the importance of diffusive transport, but because only one spacecraft was available, they had to be done on a sta- tistical basis. The interplanetary conditions, such us the solar wind velocity and density can vary significantly from event to event, therefore a statis- tical study can only reveal the general characteristics and behavior of the upstream ions, and can not give detailed information about each particular event.
2.2
Measuring the Gradient of Diffuse Ion
Partial Density
2.2.1
On the Importance of Multispacecraft Measure-
ments
The statistical studies performed by Ipavich et al. (1981) and Trattner et al. (1994) resulted in clear evidence that the ions undergo a diffusive transport. These results provide the e-folding distance of the partial ion density in front
40
Spatial Evolution of Diffuse Ion Density in Front of the Earth’s Bow Shock
of the shock, its value varying between ∼2 and ∼11 Re in the 10-67 keV energy range. These studies were based on statistical analysis of several upstream ion events, because one spacecraft can only observe the spatialand
the temporal variation of the energetic ion density. It is known that the bow shock accelerated energetic ion density is directly correlated with the density of the solar wind (Trattner et al., 1994) and is presumed that the direction of the magnetic field is also an important factor in the production of energetic ions. Even a quiet solar wind presents low-amplitude fluctuations in the plasma density and in the interplanetary magnetic field strength and direction. These fluctuations constantly and instantly influence the energetic ion production and therefore the energetic ion density itself. The energetic ion density produced at the shock might exhibit fluctuations due to changes in the solar wind plasma parameters, therefore it is vital to have measurements at different distances from the shock at the same time. This is the only way how we can assure to separate the spatial variation in partial density of the energetic ions from the temporal one.
2.2.2
Determination of the Individual Spacecraft Dis-
tance to the Shock
To measure the distance of a spacecraft to the bow shock along the mag- netic field line requires a precise calculation of the bow shock position under different interplanetary conditions. The bow shock position cannot be ob- served directly; the exact bow shock position can be observed only during spacecraft crossings. Many bow shock models have been developed and are widely used in scientific investigations since the early 1960’s (for a complete list of references see Peredo et al., 1995).
A common conclusion of many of these investigations is that the bow shock formed upstream of the Earth is a highly dynamic boundary, which
2.2 Measuring the Gradient of Diffuse Ion Partial Density 41
is controlled by steady and transient variations in solar wind parameters. Therefore choosing the appropriate bow shock model is essential in order to make a correct calculation of the bow shock position. The model needs to incorporate the response to variations in Mach number, solar wind bulk velocity and solar wind density. Taking into account the previously listed requirements, the bow shock model by Peredo et al. (1995) has been chosen. Peredo et al. (1995) analyzed a large set of bow shock crossings (about 1400 events), revealing that among the three Mach numbers (i.e., sonic ( Ms), Alfv´enic (MA) and magnetosonic (Mms) Mach numbers) MA controls the position of the bow shock much stronger than the other two. They derived a three dimensional model for the average shape and position of the bow shock under normalized solar wind conditions (see below).
The normalization takes into account variations of the bow shock position due to the solar wind dynamic pressure. Peredo et al. (1995) normalized all crossings to the average solar wind pressure of their data set according to the relation (e.g. Spreiter et al., 1966, 1968; Fairfield, 1971; Holzer and Slavin, 1978): Rnorm =Robs nobsV2 obs navgVavg2 1/6 (2.1)
where nobs,Vobs,navg, Vavg are the observed and averaged solar wind number densities and bulk speeds, respectively. Rnorm and Robs are the normalised and observed bow shock distances at subsolar point. The average quantities for the Peredo data set are navg = 7.76cm−3 and Vavg = 454.18km/s.
According to the Peredo model, the Earth’s bow shock can be repre- sented as a general second order surface which is described by the following expression:
42
Spatial Evolution of Diffuse Ion Density in Front of the Earth’s Bow Shock
F(x, y, z) =a1x2+a2y2+a3z2+a4xy+a5yz+a6xz+a7x+a8y+a9z+a10 = 0 (2.2) where the x, y and z coordinates are obtained from the GSE (Geocentric Solar Ecliptic) system via transformation, andai coefficients are functions of the Alfv´en Mach number (MA):
a1 = 0.0117−5.18×10−3MA−3.47×10−4M2A (2.3) a3 = 0.712 + 0.044MA−1.35×10−3M2A (2.4) a4 = 0.3−0.071MA+ 3.53×10−3M2A (2.5) a7 = 62.8−2.05MA+ 0.079M2A (2.6) a8 =−4.85 + 1.02MA−0.048M2A (2.7) a10 =−911.39 + 23.4MA−0.86M2A (2.8)
and a5, a6 and a9 are zero and the value of a2 is one. Note that all the coefficients are determined only by the value of MA. Horbury et al. (2001) analyzed the magnetic data during shock encounters by the four Cluster spacecraft and found the orientation of the terrestrial bow shock normal to be extremely stable, at least under steady upstream conditions. Further- more, they point out that the agreement between normals estimated from the Peredo model and those based on four spacecraft magnetic data implies that
2.2 Measuring the Gradient of Diffuse Ion Partial Density 43
even when the shock moves rapidly, the bow shock shape proves to be remark- ably stable. In conclusion, the Peredo model provides a three-dimensional bow shock model incorporating the variations of the shock surface shape un- der different Mach number conditions, which in turn according to Horbury et al. (2001) is extremely accurate and stable even under dynamically chang- ing interplanetary conditions. This makes the Peredo model reliable for the gradient study. However, like any other bow shock model, the Peredo model is based on a large data set of shock crossings. Therefore this model is able to provide only an average bow shock position. In Peredo et al. (1995) Fig- ures 3a and 3b clearly demonstrate, that the individual shock crossings are scattered around the average bow shock position with a standard deviation of ∼ ±2Re. Such a large error can not be accepted in a case study, where a precise calculation of the spacecraft distance to the bow shock is essential. In order to minimize the errors, we first identified the bow shock crossing coor- dinates by using the spacecraft data and substituted these coordinate values together with the value of the MAin Equation 2.2. This way we obtained the equation which describes the bow shock shape and position very accurately at the moment when the spacecraft is crossing it. After this we modified the bow shock position and shape according to Equation 2.1 for the whole time period of interest by using the observed solar wind velocity and number density. To summarize, in determining the spacecraft distance to the shock we used the observed bow shock surface which was continuously adjusted to the actual solar wind conditions. By knowing the actual bow shock position, the position of the spacecraft and the magnetic field direction at the space- craft, the spacecraft distance to the bow shock along the magnetic field can be easily calculated.
44
Spatial Evolution of Diffuse Ion Density in Front of the Earth’s Bow Shock